A Treatise on the Theory of Screws

306] THE GEOMETRICAL THEORY. 329 of the screws of this system will have, as its instantaneous screw, a screw of infinite pitch parallel thereto. We have thus a system of impulsive screws of the third order, and a corresponding system of instantaneous screws of the third order, the relation between each pair being quite independent of whatever particular rigid body of the group the impulsive wrench be applied to. This system of the third order taken in conjunction with the cylindroid (’?> £) will enable us to determine the total system of impulsive screws which possess the property in question. Take any screw 3, of zero pitch, passing through the centre of gravity, and any screw, </>, on the cylindroid (77, f). We know, of course, as already explained, the instantaneous screws corre- sponding to 3 and </>. Let us call them X, g, respectively. Draw the cylindroid (3, </>), and the cylindroid (X, p). The latter of these will be the locus of the instantaneous screws, corresponding to the screws on the former as impulsive screws. From the remarkable property of the two cylindroids,. so related, it follows that every impulsive screw on (3, </>) will have its corresponding instantaneous screw on (X, p) definitely fixed. This will be so, notwithstanding the arbitrary element remaining in the rigid body. From the way in which the cylindroid (3, (f>) was constructed, it is plain that the screws belonging to it are members of the system of the fifth order, formed by combinations of screws on the cylindroid (■»?, £) with screws of the special system of the third order passing through the centre of gravity. But all the screws of a five-system are reciprocal to a single screw. The five-system we are at present considering consists of the screws which are reciprocal to that single screw, of zero pitch, which passes through the centre of gravity and intersects both the screws, of zero pitch, on the impulsive cylindroid (j), %). The corresponding instantaneous screws will also form a system of the fifth order, but it will be a system of a specialized type. It will be the result of compounding all possible displacements by translation, with all possible twists about screws on the cylindroid (a, ß). The resulting system of the fifth order consists of all screws, of whatsoever pitch, which fulfil the single condition of being perpendicular to the axis of the cylindroid (a, ß). Hence we obtain the following theorem:— If an impulsive cylindroid, and the corresponding instantaneous cylin- droid, be known, we can construct, from these two cylindroids, and without any further information as to the rigid body, two systems of screws of the fifth order, such that an impulsive wrench on a given screw of one system will produce an instantaneous twist velocity about a determined screw on the other system. It is interesting to note in what way our knowledge of but two corre- sponding pairs of impulsive screws and instantaneous screws just fails to give complete information with respect to every other pair. If we take any